DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

How statement and proof provenance work

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The inverse limit is the set of compatible tuples in the Cartesian product

Definition

Let $((G_{i}), φ_{i j})$ be an inverse system of groups indexed by a directed set $I$ (A directed set and an inverse system of groups indexed by it, The Cartesian product $A \times B : = {z \in P (P (A \cup B)) : \exists a \in A \exists b \in B z = (a, b)}$ ). The inverse limit is the subset

$\underset{\leftarrow}{l i m} G_{i} : = {(g_{i})_{i \in I} \in \prod_{i \in I} G_{i} : φ_{i j} (g_{j}) = g_{i} for every i \leq j} .$

Its elements are the compatible tuples. Compatibility means exactly that all coordinates agree with the transition maps.

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)